Circumference of a circle

where r is the radius and d is the diameter of the circle, and (the Greek letter pi) is defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...).

If desired, the above circumference formula can be derived without reference to the definition of π by using some integral calculus, as follows:

The upper half of a circle centered at the origin is the graph of the function , where x runs from to . The circumference (c) of the entire circle can be represented as twice the sum of the lengths of the infinitesimal arcs that make up this half circle. The length of a single infinitesimal part of the arc can be calculated using the Pythagorean formula for the length of the hypotenuse of a right triangle with side lengths and , which gives us .